Question: Simplify; express your answer in exponential form. Assume $k\neq 0, p\neq 0$. $\dfrac{{(kp^{5})^{-3}}}{{(k^{2}p^{3})^{4}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(kp^{5})^{-3} = (k)^{-3}(p^{5})^{-3}}$ On the left, we have ${k}$ to the exponent ${-3}$ . Now ${1 \times -3 = -3}$ , so ${(k)^{-3} = k^{-3}}$ Apply the ideas above to simplify the equation. $\dfrac{{(kp^{5})^{-3}}}{{(k^{2}p^{3})^{4}}} = \dfrac{{k^{-3}p^{-15}}}{{k^{8}p^{12}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-3}p^{-15}}}{{k^{8}p^{12}}} = \dfrac{{k^{-3}}}{{k^{8}}} \cdot \dfrac{{p^{-15}}}{{p^{12}}} = k^{{-3} - {8}} \cdot p^{{-15} - {12}} = k^{-11}p^{-27}$